Worked solution · Exam P

SOA Exam P Sample Question #199 Solution

This Exam P sample reference tests Conditional Probability. The claim weights for the no-, moderate-, and heavy-lifting groups are 0.02, 0.04, and 0.02. Moderate or heavy lifting therefore accounts for 0.06 of the total 0.08 claim probability, giving 0.75, choice A.

Independent analysis by ActuaryProof · reviewed July 31, 2026

Official wording withheld. Read the problem in the official SOA sample-question PDF. This page provides only our independently written analysis.

Read the solution Open official source

Independent solution

How to solve this Conditional Probability question

Setup

Let N, M, and H denote the three lifting levels, and let C denote filing a claim.

\Pr(N)=0.40

\Pr(M)=0.50

\Pr(H)=0.10

Model

Form the joint probability of each lifting level together with a claim.

\Pr(N\cap C)=0.40(0.05)=0.02

\Pr(M\cap C)=0.50(0.08)=0.04

\Pr(H\cap C)=0.10(0.20)=0.02

Compute

Add the claim weights and normalize the two requested groups.

\Pr(C)=0.02+0.04+0.02=0.08

\Pr(M\cup H\mid C)=\frac{0.04+0.02}{0.08}=0.75

Answer

Three quarters of claimants come from the moderate- or heavy-lifting groups.

\boxed{0.75\quad\text{(A)}}

Test the method on unseen problems

General Probability Practice Workbook

30 original general probability problems with complete solutions and wrong-choice analysis. Written for Exam P, not copied from the SOA set. One-time $19 PDF.

See the cover and sample pages

Error diagnosis

Why the other letters tempt

These notes identify the calculation error associated with each wrong letter when that error is reproducible.

BThe value 0.81 would leave posterior probability 0.19 for no lifting. The verified no-lifting claim weight gives 0.02/0.08=0.25 instead.
CThe value 0.85 is approximately (0.08+0.20)/(0.05+0.08+0.20). That calculation adds conditional claim rates without weighting them by the group proportions.
DThe value 0.86 would leave posterior probability 0.14 for no lifting, understating the no-lifting joint claim weight.
EThe value 0.89 would leave only 0.11 posterior probability for no lifting. Bayes' normalization shows that the correct complement is 0.25.

Original practice · fully worked

Original variant: infer a source mix from an overall event rate

Service requests are classified as low, medium, or high complexity. Low-complexity requests form half of all requests. The high-complexity proportion is unknown, and the remainder are medium complexity. Escalation rates for the three classes are 2%, 5%, and 20%, respectively, while 5.6% of all requests are escalated. Given an escalation, calculate the probability that the request was high complexity.

A
0.028
B
0.056
C
0.140
D
0.360
E
0.500

Answer and calculation

If h is the high-complexity proportion, total probability gives 0.035+0.15h=0.056, so h=0.14. The high-complexity escalation weight is 0.14(0.20)=0.028, which is half of all escalations, choice E.

Setup

Let h be the high-complexity proportion; then the medium-complexity proportion is 0.50-h.

\Pr(L)=0.50

\Pr(M)=0.50-h

\Pr(H)=h

Model

Use the stated overall escalation probability to determine the unknown mixture proportion.

0.50(0.02)+(0.50-h)(0.05)+h(0.20)=0.056

Compute

Solve for the high-complexity share, then apply Bayes' theorem.

0.035+0.15h=0.056

h=0.14

\Pr(H\mid E)=\frac{0.14(0.20)}{0.056}=0.50

Answer

Half of escalated requests are high complexity.

\boxed{0.500\quad\text{(E)}}

Keep practicing

Next steps for Exam P General Probability

Work through the related free solutions for this topic, review the formulas you used, or keep every current Exam P solution in one searchable PDF.

More Exam P General Probability practice problems Free Exam P formula sheet Exam P study manual PDF — $49

How to solve this Conditional Probability question

Setup

Model

Compute

Answer

General Probability Practice Workbook

Why the other letters tempt

Original variant: infer a source mix from an overall event rate

Answer and calculation

Setup

Model

Compute

Answer

Related Exam P solutions

Next steps for Exam P General Probability